Group Theory (Lý Thuyết Chùm) - Nhiều Tác Giả

Thảo luận trong 'Sách tiếng nước ngoài' bắt đầu bởi Despot, 6/10/13.

  1. Despot

    Despot Lớp 11


    “1. Preface
    These notes are the outgrowth of a graduate course on Vui lòng đăng nhập hoặc đăng ký để xem link I taught at the University of Virginia in 1994. In trying to _nd a text for the course I discovered that books on Lie groups either presuppose a knowledge of di_erentiable manifolds or provide a mini-course on them at the beginning. Since my students did not have the necessary background on manifolds, I faced a dilemma: either use manifold techniques that my students were not familiar with, or else spend much of the course teaching those techniques instead of teaching Lie theory. To resolve this dilemma I chose to write my own notes using the notion of a matrix Lie group. A matrix Lie group is simply a closed subgroup of GL(n;C): Although these are often called simply \matrix groups," my terminology emphasizes that every matrix group is a Lie group.
    This approach to the subject allows me to get started quickly on Lie group theory proper, with a minimum of prerequisites. Since most of the interesting examples of Lie groups are matrix Lie groups, there is not too much loss of generality. Furthermore, the proofs of the main results are ultimately similar to standard proofs in the general setting, but with less preparation.
    Of course, there is a price to be paid and certain constructions (e.g. covering groups) that are easy in the Lie group setting are problematic in the matrix group setting. (Indeed the universal cover of a matrix Lie group need not be a matrix Lie group.) On the other hand, the matrix approach su_ces for a _rst course. Anyone planning to do research in Lie group theory certainly needs to learn the manifold approach, but even for such a person it might be helpful to start with a more concrete approach. And for those in other _elds who simply want to learn the basics of Lie group theory, this approach allows them to do so quickly.
    These notes also use an atypical approach to the theory of semisimple Lie algebras, namely one that starts with a detailed calculation of the representations of sl(3;C). My own experience was that the theory of Cartan subalgebras, roots, Weyl group, etc., was pretty di_cult to absorb all at once. I have tried, then, to motivate these constructions by showing how they are used in the representation theory of the simplest representative Lie algebra. (I also work out the case of sl(2;C); but this case does not adequately illustrate the general theory.)
    In the interests of making the notes accessible to as wide an audience as possible, I have included a very brief introduction to abstract groups, given in Chapter 1. In fact, not much of abstract group theory is needed, so the quick treatment I give should be su_cient for those who have not seen this material before. I am grateful to many who have made corrections, large and small, to the notes, including especially Tom Goebeler, Ruth Gornet, and Erdinch Tatar.”
    An Elementary Introduction to Groups and Representations
    Brian C. Hall

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